A modified compact ADI method and its extrapolation for two-dimensional fractional subdiffusion equations

Abstract

A modified Crank–Nicolson-type compact alternating direction implicit (ADI) finite difference method is proposed for a class of two-dimensional fractional subdiffusion equations with a time Riemann–Liouville fractional derivative of order $$(1-\alpha )$$ $$(0<\alpha <1)$$ . This method improves the known compact ADI methods in the sense that it is based on the L1 approximation for the fractional derivative, the truncation errors on all time levels have the same order of $$\mathcal{O}(\tau ^{2\alpha }+h_{x}^{4}+h_{y}^{4})$$ and the optimal error estimate $$\mathcal{O}(\tau ^{2\alpha }+h_{x}^{4}+h_{y}^{4})$$ can be easily obtained in the standard $$H^{1}$$ - and $$L^{2}$$ -norms and the weighted $$L^{\infty }$$ -norm. The unique solvability, unconditional stability and convergence of the resulting scheme are rigorously proved. A Richardson extrapolation algorithm is presented to increase the temporal accuracy from the order $$2\alpha $$ to the order $$\min \{1+\alpha , 4\alpha \}$$ . Numerical results demonstrate the accuracy of the modified compact ADI method and the high efficiency of the Richardson extrapolation algorithm.